Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?

• The case where $n = 2$ and the $M_j$ have non-empty branch set might already be interesting, but I am most interested in examples of higher dimension.

• One can construct branched minimal immersions in $\mathbf{R}^{n+1}$ that converge to $2 \lvert B_1^n \times \{ 0 \} \rvert$ by solving a Dirichlet problem. But these have a boundary, so it's not clear how to adapt this to the sphere.